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Existence of positive solutions of some semilinear elliptic equations with singular coefficients

Published online by Cambridge University Press:  12 July 2007

Nirmalendu Chaudhuri
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
Mythily Ramaswamy
School of Mathematics, TIFR Centre, IISc, Bangalore 560012, India


In this paper we consider the semilinear elliptic problem in a bounded domain Ω ⊆ Rn, where μ ≥ 0, 0 ≤ α ≤ 2, 2α* := 2(n − α)/(n − 2), f : Ω → R+ is measurable, f > 0 a.e, having a lower-order singularity than |x|-2 at the origin, and g : R → R is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions Ip for which the embedding is compact. When p = 2, α = 2, fI2 and 0 ≤ μ < (½(n − 2))2, we prove that the linear problem has -discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of I2, the first eigenvalue goes to a positive number as μ approaches (½(n − 2))2. Furthermore, when g is superlinear, we show that for the same subclass of I2, the functional corresponding to the differential equation satisfies the Palais-Smale condition if α = 2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0 ≤ α < 2.

Research Article
Copyright © Royal Society of Edinburgh 2001

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